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You probably have some idea of what we mean when we say ``probability,'' but here's a definition to clarify:
\begin{proc}{What is Probability?}
\paragraph{Probability:} \text{}
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\paragraph{Probability of something occurring:} \text{}
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\end{proc}
\subsection{Vocabulary}
\begin{itemize}
\item \textbf{Outcome:} \text{}
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\item \textbf{Sample Space:} \text{}
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\begin{enumerate}[(a)]
\item\text{}
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\item \text{}
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\item \text{}
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\end{enumerate}
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\item \textbf{Event:} \text{}
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The probability of an event $A$ is written $P(A)$, or we could write $P($rolling a $4)$ or $P(4)$, if that is clear enough in context.
\end{itemize}
\begin{example}{Two siblings}
Consider randomly selecting a family with 2 children where the order in which different gender siblings are born is significant. That is, a family with a younger girl and an older boy is different from a family with an older girl and a younger boy. What would the sample space look like? \\
\marginnote{\bfseries Solution} If we let G denote a girl, B denote a boy, then we have the following:
\[ S = \left\{ GG, BB, GB, BG \right\} \]
This notation represents families with 2 girls, 2 boys, an older girl and a younger boy, an older boy and a younger girl.
\end{example}
\begin{example}{Three Siblings}
What would the sample space $S$ look like if we considered a family with 3 children? Remember, the order of children born is significant. \\
Now there are 8 possibilities:
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\end{example}
\begin{example}{Tossing a coin and rolling a die}
Suppose we toss a fair coin and then roll a six-sided die once. Describe the sample space $S$. \\
\marginnote{\bfseries Solution \\ \includegraphics[height = .3in]{dice}} Let $T$ denote Tails, and $H$ denote Heads. Then
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\end{example}
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\paragraph{Note:} in the definition of probability, we said that probability is a \textbf{proportion}.\\
\begin{proc}{Probability}
The basic rules of probability are:
\begin{enumerate}
\item \text{}
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\item \text{}
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\item \text{}
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\item \text{}
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\end{enumerate}
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Often we use percentages to represent probabilities. For example, a weather forecast might say that there is 85\% chance of rain in Frederick tomorrow. Or there is 67\% chance that the Baltimore Orioles will win their next series. Or a particular poker player has a 35\% chance of winning the game with his current hand. As you might have already guessed, 100\% chance corresponds to 1, and 0\% corresponds to 0.
\subsection{Theoretical Probability}
There are two types of probability: \textbf{theoretical} and \textbf{empirical}. Theoretical probability is used when the set of all equally-likely outcomes is known. To compute the theoretical probability of an event $A$, denoted $P(A)$, we use the formula below:
\begin{formula}{Theoretical probability}
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\end{formula}
This makes sense with the definition of probability, namely that it is the proportion of times we would expect $E$ to occur if we repeated the experiment many times. This proportion comes from dividing the number of possibilities that correspond to $E$ by the total number of possibilities there are.
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In the example below, one could probably find the probability by intuition, but it's good to know how to apply the formula, even in what seems to be a simple experiment.
\begin{example}{Rolling a die}
Assume you are rolling a fair six-sided die. What is the probability of rolling an odd number?\\
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\end{example}
\begin{example}{Three Siblings}
Consider the earlier example about a family with three children. Remember, the sample space looked like
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Let's find the probability of a few combinations of kids:
\begin{enumerate}[(a)]
\item $P($three girls$) =$ \text{}
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\item $P($at least two boys$) =$ \text{}
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\item $P($exactly one girl$) =$ \text{}
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\item $P($youngest is a boy$) =$ \text{}
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\item $P($oldest and youngest same$) =$ \text{}
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\end{enumerate}
\end{example}
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In the next example, it is not necessary to list all possible outcomes of an experiment. However, if you are not familiar with a standard deck of 52 cards, the diagram below should be helpful.
\begin{center}
\includegraphics[width=0.8\textwidth]{playingcards}
\end{center}
\begin{example}{Drawing a card}
Suppose you draw one card from a standard 52-card deck. What is the probability of drawing an Ace? \\
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\end{example}
\begin{example}{Drawing Another Card}
\begin{enumerate}[(a)]
\item When drawing a card from a standard 52-card deck, what is the probability of drawing a face card? \emph{Face cards include Jacks, Queens, and Kings}.
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\item What is the probability of drawing the King of Hearts?
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\end{enumerate}
\end{example}
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\begin{example}{Cookie Jar}
Lisa's cookie jar contains the following: 5 peanut butter, 10 oatmeal raisin, 12 chocolate chip, and 8 sugar cookies. If Lisa selects one cookie, what is the probability she gets a peanut butter cookie? \\
\marginnote{\includegraphics[height = .8in]{cookies} } \text{}
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\end{example}
\subsection{Empirical Probability}
As long as we can list--or at least count--the sample space and the number of outcomes that correspond to our event, we can calculate basic probabilities by dividing, as we have done so far. But there are many situations where this isn't feasible.
For instance, take the example of a batter coming to the plate in a baseball game. There's no way to even begin to list all the possible outcomes that could occur, much less count how many of them correspond to the batter getting a hit. We'd still like to be able to estimate the likelihood of the batter getting a hit during this at-bat, though. Just as sports fan do, then, we turn to this batter's previous performance; if he's gotten a hit in 200 of his last 1000 at-bats, we assume that the probability of a hit this time is $\frac{200}{1000}=0.200$.
Empirical probability is used when we observe the number of occurrences of an event. It is used to calculate probabilities based on the \emph{real data} that we observed and collected. To compute the empirical probability of an event $E$, denoted $P(E)$, we use the formula below:
\begin{formula}{Empirical probability}
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\end{formula}
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This can also be used to answer questions about sampling randomly from a population if we know the breakdown of the group.
\begin{example}{FCC students}
Consider the following information about FCC students' enrollment:
\begin{center}
\begin{tabular}{|c|c|}
\hline
Gender & Enrollment \\ \hline
Female & 3653 \\ \hline
Male & 2580 \\ \hline
\end{tabular}
\end{center}
If one person is randomly selected from all students at FCC, what is the probability of selecting a male?\\
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\end{example}
The next example contains a two-way table, often referred to as \emph{contingency} table, which breaks down information about a group based on two criteria. For example, the table below breaks down a group of 130 FCC students based on gender and which hand is their dominant hand:
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Gender & Right-handed & Left-handed \\ \hline
Female & 58 & 13\\ \hline
Male & 47 & 12 \\ \hline
\end{tabular}
\end{center}
In order to use this to calculate probabilities if we randomly select someone from the group, we need to calculate totals for each category: the number of males, the number of females, the number of left-handed people, and the number of right-handed people. This is done by simply summing each row and column; if we do that, we obtain the completed table below.
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Gender & Right-handed & Left-handed & \textbf{Total} \\ \hline
Female & 58 & 13 & 71\\ \hline
Male & 47 & 12 & 59 \\ \hline
\textbf{Total} & 105 & 25 & 130 \\ \hline
\end{tabular}
\end{center}
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\begin{example}{FCC students}
Consider the following information about a group of 130 FCC students:
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
Gender & Right-handed & Left-handed & \textbf{Total} \\ \hline
Female & 58 & 13 & 71\\ \hline
Male & 47 & 12 & 59 \\ \hline
\textbf{Total} & 105 & 25 & 130 \\ \hline
\end{tabular}
\end{center}
\begin{enumerate}[(a)]
\item If one person is randomly selected from the group, what is the probability this student is left-handed? \\
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\item Find the probability of selecting a female student.\\
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\end{enumerate}
\end{example}
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